A Fourier wavelet series solution of partial differential equation through the separation of variables method

This paper investigates the modal analysis of vibration of Euler-Bernoulli beam subjected to concentrated load. The governing partial differential equation was analysed to determine the behaviour of the system under consideration. The series solution and numerical methods were used to solve the governing partial differential equation. The results revealed that the amplitude increases as the length of the beam increases. It was also found that the response amplitude increases as the foundation increases at fixed length of the beam.

Download Full-text

Explicit Solution for Cylindrical Heat Conduction

American Journal of Undergraduate Research ◽

10.33697/ajur.2016.020 ◽

2016 ◽

Vol 13 (2) ◽

Cited By ~ 1

Author(s):

Kaitlyn Parsons ◽

Tyler Reichanadter ◽

Andi Vicksman ◽

Harvey Segur

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Heat Conduction ◽

Boundary Conditions ◽

Heat Equation ◽

Analytical Method ◽

Explicit Solution ◽

Separation Of Variables ◽

Partial Differential ◽

Forcing Function

The heat equation is a partial differential equation that elegantly describes heat conduction or other diffusive processes. Primary methods for solving this equation require time-independent boundary conditions. In reality this assumption rarely has any validity. Therefore it is necessary to construct an analytical method by which to handle the heat equation with time-variant boundary conditions. This paper analyzes a physical system in which a solid brass cylinder experiences heat flow from the central axis to a heat sink along its outer rim. In particular, the partial differential equation is transformed such that its boundary conditions are zero which creates a forcing function in the transform PDE. This transformation constructs a Green’s function, which admits the use of variation of parameters to find the explicit solution. Experimental results verify the success of this analytical method. KEYWORDS: Heat Equation; Bessel-Fourier Decomposition; Cylindrical; Time-dependent Boundary Conditions; Orthogonality; Partial Differential Equation; Separation of Variables; Green’s Functions

Download Full-text

The Use of Homotopy Analysis Method to Solve the Time-Dependent Nonlinear Eikonal Partial Differential Equation

Zeitschrift für Naturforschung A ◽

10.1515/zna-2011-0501 ◽

2011 ◽

Vol 66 (5) ◽

pp. 259-271 ◽

Cited By ~ 6

Author(s):

Mehdi Dehghan ◽

Rezvan Salehi

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Homotopy Analysis Method ◽

Series Solution ◽

Research Work ◽

Time Dependent ◽

Test Problems ◽

Analysis Method ◽

Partial Differential ◽

Homotopy Analysis

In this research work a time-dependent partial differential equation which has several important applications in science and engineering is investigated and a method is proposed to find its solution. In the current paper, the homotopy analysis method (HAM) is developed to solve the eikonal equation. The homotopy analysis method is one of the most effective methods to obtain series solution. HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of a series solution. Furthermore, this method does not require any discretization, linearization or small perturbation and therefore reduces the numerical computation a lot. Some test problems are given to demonstrate the validity and applicability of the presented technique.

Download Full-text

On the problem of convergence of series solution of non-linear fractional partial differential equation

10.1063/1.4990326 ◽

2017 ◽

Cited By ~ 2

Author(s):

Prince Singh ◽

Dinkar Sharma

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Series Solution ◽

Fractional Partial Differential Equation ◽

Partial Differential ◽

Convergence Of Series ◽

Non Linear

Download Full-text

Mass transfer in droplets determined by analog computation

SIMULATION ◽

10.1177/003754977001500601 ◽

1970 ◽

Vol 15 (6) ◽

pp. 241-248 ◽

Cited By ~ 1

Author(s):

R.M. Wellek ◽

J.T. Kuo ◽

R.C. Waggoner

Keyword(s):

Differential Equation ◽

Mass Transfer ◽

Partial Differential Equation ◽

Separation Of Variables ◽

Analog Computer ◽

Internal Circulation ◽

Eigenvalues And Eigenfunctions ◽

Partial Differential ◽

Sturm Liouville ◽

Single Droplets

A mechanism describing the rate of mass transfer to single droplets with a special type of internal circulation is described by a model consisting of a partial differential equation with two independent variables. A Sturm-Liouville system is obtained when the partial differential equation is transformed into a set of ordinary differential equations by the separation-of-variables technique. The eigenvalues and eigenfunctions which determine the solution to this system are obtained by a two variable search procedure on an iterative-analog computer.

Download Full-text

The Application of MATLAB in Classical Electrostatics Boundary-Value Problems

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.378.602 ◽

2013 ◽

Vol 378 ◽

pp. 602-608

Author(s):

Fu Jian Zong ◽

Jin Ma

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Boundary Value Problems ◽

Potential Function ◽

Laplace Equation ◽

Electrostatic Field ◽

Computer Image ◽

Separation Of Variables ◽

Boundary Value ◽

Partial Differential

In this paper we introduce the use of a computer image and the Partial Differential Equation (PDE) Toolbox in MATLAB, and discuss the electrostatic field, the potential function and the solution of the Laplace equation by separation of variables and the PDE toolbox. It is convenient to figure out the classical electrostatics problem with MATLAB.

Download Full-text

Accelerated HPSTM: An efficient semi-analytical technique for the solution of nonlinear PDE’s

Nonlinear Engineering ◽

10.1515/nleng-2020-0019 ◽

2020 ◽

Vol 9 (1) ◽

pp. 329-337

Author(s):

Deepak Grover ◽

Dinkar Sharma ◽

Prince Singh

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Series Solution ◽

Nonlinear Partial Differential Equation ◽

Transformation Method ◽

Proportional Delay ◽

Approximate Analytic Solution ◽

Partial Differential ◽

Analytical Technique ◽

Homotopy Perturbation

AbstractIn this paper a novel technique i.e. accelerated homotopy perturbation Sumudu transformation method (AHPSTM), which is a hybrid of accelerated homotopy perturbation method and Sumudu transformation to obtain an approximate analytic solution of nonlinear partial differential equation (PDE) with proportional delay, is used. This approach is based on the new form of calculating He’s polynomial, which accelerates the convergence of the series solution. The series solutions obtained from the proposed method are found to converge rapidly to exact solution. In order to affirm the effectiveness and legitimacy of proposed method, the proposed technique is implemented on nonlinear partial differential equation (PDE) with proportional delay. The condition of convergence of series solution is analyzed. Moreover, statistical analysis has been performed to analyze the outcome acquired by AHPSTM and other semi-analytic techniques.

Download Full-text

Convergence and Error Analysis of Series Solution of Nonlinear Partial Differential Equation

Nonlinear Engineering ◽

10.1515/nleng-2017-0113 ◽

2018 ◽

Vol 7 (4) ◽

pp. 303-308 ◽

Cited By ~ 5

Author(s):

Prince Singh ◽

Dinkar Sharma

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Error Analysis ◽

Series Solution ◽

Homotopy Perturbation Method ◽

Nonlinear Partial Differential Equation ◽

Partial Differential ◽

Uniqueness Of The Solution ◽

Established Fact ◽

Sumudu Transforms

Abstract A hybrid method of Sumudu transforms and homotopy perturbation method (HPM) is used to solve nonlinear partial differential equation. Here the nonlinear terms are handled with He’s polynomial to obtain the series solution. But, for the authenticity of the obtained solution, the condition of convergence and uniqueness of the solution is derived. The facts are obtained in reference to convergence and error analysis of this solution. Finally, the established fact is supported by finding solution of two well known equations Newell-Whitehead Segel and Fisher’s equation

Download Full-text